if-a-b-gt-0-and-a-2-b-2-5-3-find-a-2-b-2-ab-

Question Number 89991 by jagoll last updated on 20/Apr/20

if a, b > 0 and \frac{a^{2}}{b^{2}} = \frac{5}{3}

find \frac{a^{2} + b^{2}}{ab}

Commented by john santu last updated on 20/Apr/20

a b = \sqrt{a^{2} b^{2}}, s i n c e a, b > 0

a^{2} = \frac{5}{3} b^{2} \Rightarrow a b = \sqrt{\frac{5 b^{4}}{3}} = \frac{b^{2} \sqrt{15}}{3}

t h e n \frac{a^{2} + b^{2}}{a b} = \frac{\frac{8 b^{2}}{3}}{\frac{b^{2} \sqrt{15}}{3}}

= \frac{8}{\sqrt{15}}

Commented by mr W last updated on 20/Apr/20

{(\frac{a}{b})}^{2} = \frac{5}{3}

\frac{a}{b} = \frac{\sqrt{5}}{\sqrt{3}}

\frac{a^{2} + b^{2}}{a b} = \frac{a}{b} + \frac{b}{a} = \frac{\sqrt{5}}{\sqrt{3}} + \frac{\sqrt{3}}{\sqrt{5}} = \frac{8}{\sqrt{15}} = \frac{8 \sqrt{15}}{15}

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